# 5.3 - MATH 1081  Wednesday, March 2   ...

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Unformatted text preview: MATH 1081  Wednesday, March 2    Chapter 5 – Section 3    HIGHER DERIVATIVES, CONCAVITY, AND  THE SECOND DERIVATIVE TEST    Homework #7 (due 3/7)             Section 5.2 #30, 36, 46, 56  Section 5.3 #30, 44, 52, 78, 82, 94  Clicker Check­In:  Choose any letter to check in now.  To recap what we have been doing in the last two sections, we  are using the sign of the derivative to determine where a  function is increasing and where it is decreasing.  More  precisely, on intervals where  f ' ( x ) > 0 , the function  f  is  increasing and on intervals where  f ' ( x ) < 0 , the function  f  is  decreasing.    We can then use that information to determine the location of  relative extrema, looking at points in the domain where the  function  f  changes direction.    This analysis gives us a rough idea of what the graph of the  function must look like.  However, this analysis alone leaves  some information about the shape of the graph unknown.  For  example, just looking at the sign of the derivative, there is no  difference between the two functions shown here.      However, there clearly is a difference between the shapes of  these functions.    Consider these two curves, both of which are increasing on the  interval ( 0, 5 ) .             How can we describe the difference between the shapes of  these curves?  What can we say about the tangent lines in each case? Now consider these two curves, both of which are decreasing  on the interval ( 0, 5 ) .            How can we describe the difference between the shapes of  these curves?  What can we say about the tangent lines in each case?  Definition: Concavity  A function is concave up on an interval ( a, b ) if the graph of  the function lies above its tangent line at each point in the  interval.  A function is concave down on an interval ( a, b )  if the graph  of the function lies below its tangent line at each point in  the interval.  A point at which the graph changes concavity is called an  inflection point.    From the discussion on the last two slides, “concave up”  corresponds to a curve where the slopes of the tangent lines  are increasing, whereas, “concave down” corresponds to a  curve where the slopes of the tangent lines are decreasing.   From Section 5.1, we have a method to determine where a  function is increasing and where something is decreasing…we  look at its derivative.    So, to measure the concavity of a function  f , we need to  analyze the derivative of its “slope function”.  But the “slope  function” of  f  is  f ' ( x ) .  This means that we want to analyze the  derivative of the derivative of  f , which we call the second  derivative of  f .    Notation: Common notations for the second derivative are    d2y 2         Dx ⎡ f ( x ) ⎤   f '' ( x )        ⎣ ⎦ dx 2 Example:  Find the second derivative for the function.          1.  f ( x ) = −7 x 2 + 9 x + 3          8       2.   g ( x ) = 5 x 4 / 3 −   x         5x − x2       3.   y = e − 2 e   Suppose  s ( t )  gives us the position of an object at time  t .    What physical interpretation does the derivative  s ' ( t )  have?      What physical interpretation does the second derivative  s '' ( t )   then have? Given an algebraic expression for a function  f ( x ) , we must  look at the sign of the second derivative  f '' ( x )  (because that  will tell us if  f ' ( x )  is increasing or decreasing) to determine  concavity.    If both  f '  and  f ''  exist at all points in an interval ( a, b )  and     • f '' ( x ) > 0  at all  x  in that interval, then  f  is concave upward  on ( a, b ) .  • f '' ( x ) < 0  at all  x  in that interval, then  f  is concave  downward on ( a, b ) .    A point in the domain of  f  where the concavity (sign of  f '' )  changes is an inflection point.  Example:  Determine the intervals on which the function is   concave upward and the intervals on which the     function is concave downward and identify any     inflection points.          1.   f ( x ) = −7 x 2 + 9 x + 3      5       2.   g ( x ) =   x−2       − x2       3.   h ( x ) = 2 e   One useful application of the point of inflection is to describe  the point of diminishing return.    Suppose you begin a new ad campaign.  At first, there is an  explosion in your customer base…the rate of increase of the  number of customers is high.    Eventually, the rate at which new customers appear will slow  down.  You are still adding new customers (so the first  derivative is still positive), but at a slower rate (so the second  derivative turns negative).     Clicker Check­out:  Choose any letter to check out now.      Tomorrow in recitation:  Workshop #4    Next time – Section 6.1  ...
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